Z-scores: relative to which group?
A z-score tells you how many reference SDs a value lies above or below the reference mean. It has no units.
z = (value - reference mean) / reference SD. It is unitless and says how many reference SDs the value is above or below the reference mean. Positive does not automatically mean good. Computing z does not require a normal distribution; turning z into a normal-model percentile does.
For score 99, cohort mean 65.86 and SD 10.94, z is about 3.03. The same score can have different z-values against another programme because the reference mean and SD differ. State the population and the SD convention.
If all values are identical, the reference SD is zero and the usual z formula is undefined. Standardising arbitrary programme codes or IDs gives numbers without a meaningful quantitative interpretation. A standardised column has mean approximately zero and SD one when evaluated on the same reference data with the same convention, apart from numerical precision.
Distance versus rank: in the lab's minutes [1,2,...,9,M], M ranges from 46 to 66. The mean is (45+M)/10, greater than 9, so 9 is below the mean. Nevertheless 9 of the 10 observations are at or below 9: it is at the 90th percentile by the stated empirical convention. The large tail value separates mean position from rank.
Specification versus statistical unusualness: bottle reference mean 100 mL and SD 2 mL, with an agreed acceptance interval 94-106 mL. A 104 mL bottle has z=2 and is accepted; a 108 mL bottle has z=4 and is rejected. Acceptance comes from the specification, not from an automatic z cutoff. Preserve the rejected bottle's measurement for process analysis.
Sources: W2 PDF pp. 2-60; L2 Sections 1-8. The personalised Lab 2 has 240 synthetic students, unlike the 1,200-student lecture dataset. Do not mix their numbers.
The same raw score gets different z-scores when the reference changes. This lab uses sample SD (ddof=1); the Week 4 prediction scaler uses training-only population-style SD (ddof=0).
Practice problems
Supplementary study questions on the lecture concepts — not verbatim exam questions. Each answer counts toward spaced review.
1. Conceptual understanding
Definitions and distinctions.
A value 87 has reference mean 75 and SD 6. What is z?
Does a z-score alone determine a percentile?
2. Application
Apply the concept to a case or calculation.
Value 60, mean 70, SD 5 gives which z-score?
What happens to the ordinary z formula when reference SD is zero?
3. Interpretation and limitations
Evaluate assumptions and justify conclusions.
A waiting time has z=2. Does that automatically mean a good outcome?
A bottle is within its documented acceptance interval but has z=2. What decides acceptance?
Explain it in writing
Self-checked, not marked. Write a few sentences, then compare with the example answer. Your writing is saved as you type.
Show an example answer
Yes. In a strongly right-skewed list, a very large tail value can pull the mean above most observations. A value can then be below the mean but above many observations.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?
Show an example answer
The reference groups can have different means or SDs. Always name the reference rather than treating z as a property of the score alone.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?